280
9. E A R T H Q U A K E S A N D S O U R C E T H E O R Y
in which case ηR approaches 1 and the fracture energy, EG , goes to zero as the
rupture velocity approaches the shear wave velocity. For about 30 earthquakes of
6.6 < MW < 8.3, Venkataraman and Kanamori (2004) obtained radiation efficiency
estimates generally between 0.25 and 1.0. One class of earthquakes that appear to
have ηR < 0.25 are tsunami earthquakes, which involve slow rupture and generate
large tsunamis relative to their moment.
The radiation efficiency should not be confused with the seismic efficiency, η,
defined as the fraction of the total energy that is radiated into seismic waves:
η=
ER
µẽ
µER
ER
=
=
.
=
E
σM0
σ
σDA
(9.66)
The seismic efficiency is more difficult to estimate than the radiation efficiency
because it depends upon the poorly constrained absolute stress level on the fault.
In the extreme case where we assume that the earthquake relieves all of the stress
on the fault, then σ2 = 0 and we say that the stress drop is total. In this case, EF = 0
and we have
Emin = 12 σDA =
σ
M0 .
2µ
(9.67)
This represents the minimum amount of energy release for an earthquake with a
given stress drop and moment.
The theories that describe how slip on a fault initiates, propagates, and comes to
a halt can be very complicated, even for idealized models with uniform pre-stress
and elastic properties. Much of the recent work in this area has involved theory and
observations of rate and state friction (e.g., Dieterich, 1994) in which the frictional
properties are time and slip dependent. Because these models vary in their behavior
and it is likely that real earthquakes span a range of different rupture properties, it is
important to keep in mind the distinction between parameters that are more-or-less
directly estimated (e.g., moment, geodetically-determined static stress drop, and radiated energy) and those that depend upon modeling assumptions (e.g., Brune-type
and Orowan stress drops) and thus are not truly independent measurements. For example, it would make little sense to use (9.60) to estimate EG if both σ and ER are
derived from fitting the observed body-wave spectra to the same theoretical model.
9.7 Earthquake magnitude
For historical reasons the most well-known measure of earthquake size is the earthquake magnitude. There are now several different types of magnitude scales, but
all are related to the largest amplitude that is recorded on a seismogram. This is one
9.7 E A R T H Q U A K E M A G N I T U D E
281
of the easiest things to measure and is one reason for the continued popularity of
magnitude scales. A recent comprehensive review of magnitude scales is contained
in Utsu (2002b).
In the 1930s, Charles Richter introduced what is now called the local magnitude
ML . This was determined by measuring the largest amplitude A recorded on a
standard instrument, the Wood–Anderson seismograph. Richter noticed that plots
of log A versus epicentral distance for different earthquakes generally exhibited a
similar decay rate (Fig. 9.23). This suggested that a distance-independent measure
of earthquake size could be provided by the offset in log A from a reference event
at the same range,
ML = log10 A(X) − log10 A0 (X),
(9.68)
where A0 is the amplitude of the reference event and X is the epicentral distance.
At each seismic station, a value of ML may be obtained from the measured amplitude A and the value of log10 A0 at the appropriate source–receiver distance
(Richter made a table of log10 A0 at different ranges). From the table of values of
A0 (X), an approximate empirical formula has been derived (e.g., Bullen and Bolt,
1985):
ML = log10 A + 2.56 log10 X − 1.67,
(9.69)
where A is the displacement amplitude in microns (10−6 m) and X is in kilometers.
The formula is valid for 10 < X < 600 km. For the Wood–Anderson torsion instrument the largest amplitude generally comes from the S-wave arrival. Individual
log(A)
event 1
event 2
event 3
X
Figure 9.23 Different earthquakes are observed to have a similar falloff in log(amplitude) with
distance.
282
9. E A R T H Q U A K E S A N D S O U R C E T H E O R Y
estimates of ML will exhibit some scatter owing to directivity, radiation pattern,
focusing, and other effects. However, a stable estimate can generally be obtained
by averaging the results from different stations.
Richter defined a fairly small reference event so that the magnitudes of all but
the tiniest earthquakes are positive. Events below about ML = 3 are generally
not felt. Significant damage to structures in California begins to occur at about
ML = 5.5. The magnitude scale is logarithmic to account for the wide range in
observed amplitudes. A ML = 6.0 event implies a recorded amplitude 100 times
greater than a ML = 4.0 event.
The Richter magnitude scale provided a practical method of quickly determining
the relative size of different events in California. Since the dominant period of the
Wood–Anderson intrument (0.8 s) is close to that of many structures, the ML scale
has proven especially useful in engineering seismology. The local magnitude scale
is also important because all subsequent magnitude scales have been tied to it.
However, the portability of ML is limited since it is based upon an amplitude
versus range relationship that was defined specifically for southern California, and
it depends on an intrument that is now rarely used. Caltech and Berkeley kept some
Wood–Anderson seismographs operating into the 1990s just to maintain continuity
of the magnitude scale. However, these venerable instruments have now been retired
since the Wood–Anderson response can be simulated through suitable filtering of
modern broadband data. Related to ML for local earthquakes is the coda magnitude
(e.g., Suteau and Whitcomb, 1979), which is derived from the amplitude of the
scattered waves or coda that follow the direct P and S arrivals, and which has been
calibrated to agree with the local magnitude scale. In many cases, coda magnitudes
are more stable than ML because the scattered energy that makes up the coda
waves averages out spatial variations and provides a more uniform coverage of the
radiation pattern.
A general magnitude scale used for global seismology is the body-wave magnitude, which is defined as
mb = log10 (A/T) + Q(h, ),
(9.70)
where A is the ground displacement in microns, T is the dominant period of the
measured waves, is the epicentral distance in degrees, and Q is an empirical
function of range and event depth h (e.g., Veith and Clawson, 1972). The Q function
includes the details of the average amplitude versus epicentral distance and source
depth behavior of the Earth. The measurement is normally made on the first few
cycles of the P-wave arrival on short-period vertical-component intruments, for
which the dominant wave period is usually about 1 s. As with the local magnitude
scale, mb estimates for the same event will vary between stations, with scatter of up
9.7 E A R T H Q U A K E M A G N I T U D E
283
to about ±0.3. This is due to radiation pattern, directivity, and local station effects.
A station correction term is often used to account for stations that consistently give
higher or lower mb values.
Another global seismology scale is the surface wave magnitude, which may be
defined as
MS = log10 (A/T) + 1.66 log10 + 3.3
(9.71)
for Rayleigh wave measurements on vertical instruments. Since the strongest
Rayleigh wave arrivals are generally at a period of 20 s, this expression is often
written as
MS = log10 A20 + 1.66 log10 + 2.0.
(9.72)
Note that this equation is applicable only to shallow events; surface wave amplitudes
are greatly reduced for deep events.
The mb and MS scales were designed to agree with the ML scale for local events
in California. However, it is not possible to align the scales for all size events. This
is because the magnitude scales are obtained at different periods and the frequency
content of events changes as a function of event size. Consider the examples in
the previous sections, in which the source spectrum falls as f −2 above a certain
corner frequency. The corner frequency fc generally moves to lower frequencies
for larger events. If we assume stress drop is constant, then the fault dimension
1/3
and corresponding rupture duration will scale approximately as M0 . The corner
−1/3
frequency is inversely proportional to the rupture duration and will scale as M0 .
In this case the position of the corner will fall off as f −3 (Fig. 9.24).
At frequencies below fc there is a linear relationship between magnitude (log10 of
the measured amplitude) and moment. However, at higher frequencies this linearity
breaks down and the magnitude scale does not fully keep up with the increasing
size of the events. This phenomenon is called magnitude saturation. At a given
measurement frequency (e.g., 1 Hz for mb ) this begins to occur when the event
size becomes large enough to move the corner frequency below the measurement
frequency. Of course, not all earthquake spectra fall off exactly at f −2 but any degree
of falloff will lead to some saturation of the magnitude scale. Another contributing
factor to magnitude saturation can be the fixed window length used to measure the
amplitudes, which may not be long enough to capture the true amplitude of larger
events (e.g., Houston and Kanamori, 1986). Observed mb values begin to saturate
at about mb = 5.5 and MS values (measured at longer period) at about MS = 8. For
this reason it is rare for mb to exceed 7 or for MS to exceed 8.5, even for extremely
large events.
9. E A R T H Q U A K E S A N D S O U R C E T H E O R Y
284
1021
MW = 8
mb
MS
A( f ) (scaled by M0, N m)
7
1019
6
1017
5
4
1015
3
1013
2
1011
MW = 1
0.01
0.1
1
10
100
Frequency (Hz)
Figure 9.24 For larger events, the corner in the source spectrum moves to lower frequencies,
reducing the observed amplitude increase at the fixed frequencies used to estimate MS and mb .
The saturation of the mb and ML scales for large events helped motivate development of the moment magnitude MW by Kanamori (1977) and Hanks and Kanamori
(1979). The moment magnitude is defined as
(9.73)
MW = 23 log10 M0 − 9.1 ,
where M0 is the moment measured in N m (for M0 in dyne-cm, replace the 9.1
with 16.1)2 . The moment magnitude is derived entirely from the moment, with a
scaling such that MW is in approximate agreement with MS for many events. The
advantage of the MW scale is that it is clearly related to a physical property of
the source and it does not saturate for even the largest earthquakes. It is simply
another way to express the moment, which provides units that are easier to quickly
comprehend than M0 numbers like 8.2 × 1019 N m.
However, (9.73) can be misused if it is naively applied to estimate M0 from an
earthquake magnitude. This is because the ML , mb , and MS magnitudes exhibit
considerable scatter among events of the same moment and even their average
values do not agree with MW over the full range of event sizes. To see this, consider
MS measurements and the self-similar ω−2 source spectra plotted in Figure 9.24.
2 A minor source of confusion has existed in definitions of M , arising from a lack of precision in the final term.
W
The original Hanks and Kanamori paper defined moment magnitude as 2/3 log M0 − 10.7 (dyne-cm). However,
many authors, including Aki and Richards (2002), use log M0 = 1.5MW + 16.1, which is slightly different
(note that 1.5 × 10.7 = 16.05). Additional slight precision loss can occur in translating from dyne-cm to N m
because of the 2/3 factor. Here we use the Aki and Richards definition of MW .
9.7 E A R T H Q U A K E M A G N I T U D E
285
For small earthquakes (MW < 6), the measured amplitude at 20 s will scale linearly
with moment and thus MS ∝ log10 M0 . For larger earthquakes (MW > 8), the result
of the f −3 corner frequency falloff and f −2 high-frequency spectral falloff is that
MS ∝ 31 log10 M0 . MW was defined to agree with MS mainly for events between
M 6 and 8, where a slope of 2/3 is approximately correct. Thus, we should expect
MS to underpredict MW at both small and large magnitudes. A similar phenomenon
should occur for mb measurements, but shifted to smaller earthquakes because of
the higher frequency of the mb observations.
This is illustrated in Figure 9.25, which plots as gray corridors the distribution
of MS and mb measurements from the US Geological Survey’s Preliminary Determination of Epicenter (PDE) catalog as a function of M0 values for the same
earthquakes from the Global CMT catalog between 1976 and 2005. For reference,
the straight line shows MW values from (9.73), which has a slope of 2/3 on the log
plot. MS agrees approximately with MW at magnitudes between about 6.5 and 7.5,
but underpredicts MW outside of this interval. The mb values agree with MW near
magnitude 5, but increasingly underpredict MW at larger magnitudes. The average mb and MS values agree exactly only near magnitude 5.5, where they slightly
9
Magnitude or log10(A)
8
1
MS
M0
2/3
7
M0
6
M0
1s
mb
1/3
5
4
MW
20 s
3
1012
14
10
16
10
18
10
20
10
22
10
Moment (N m)
Figure 9.25 Magnitude as a function of moment, M0 , for mb , MS , and MW , compared to
predictions of log(amplitude) for an ω−2 source model at periods of 1 and 20 s.The gray
corridors show mb and MS values (± one standard deviation) from the USGS PDE catalog versus
M0 from the Global CMT catalog, compared to the definition of MW (straight solid line).The
dashed lines are predicted P-wave amplitudes for the Madariaga (1976) source model, assuming
a stress drop of 3 MPa.
286
9. E A R T H Q U A K E S A N D S O U R C E T H E O R Y
underpredict the MW value of 5.8. This behavior can be explained nicely with an
ω−2 source model, assuming constant stress drop. This model predicts that log10 (A)
2/3
will vary as M0 for small events, as M0 for events with corner frequencies near
1/3
the observation frequency, and as M0 for larger events. Predicted log10 amplitudes at the mb measurement period of 1 s and the MS measurement period of 20 s
exhibit similar behavior to the data.
Figure 9.25 makes clear how the 2/3 factor in the definition of MW serves to
make MW values comparable on average to standard magnitudes over a fairly wide
range of event sizes, provided mb is used for earthquakes less than about 5.5 and MS
is used for larger events. However, this correspondence increasingly fails outside
of 4 < MW < 8. Moment is not routinely estimated for earthquakes below about
MW = 4, which are only recorded locally. However, because ML is computed at
a similar frequency to mb , we should not expect the MW and ML scales to agree
very well for small earthquakes. Results for the slope of ML versus log10 (M0 ) have
varied among different studies, but a systematic analysis of over 60 000 southern
California earthquakes of 1 ≤ ML ≤ 3 by Shearer et al. (2006) gave a best-fitting
slope of 0.96, close to the unit slope predicted from Figure 9.25.
The various seismic magnitude scales are important because they are ingrained
in the history and practice of seismology. But if earthquake size is to be quantified with a single number, it is far better to use the moment because it is directly
related to a fundamental physical property of the source, which can also be used
in geodetic studies of earthquakes and comparisons to long-term geological slip
rates. Although it is possible to understand the average behavior of the various
magnitude scales with respect to moment, it should be remembered that ML , mb ,
and MS measurements exhibit large scatter among individual events of the same
moment. These differences are presumably not random and may reflect variations
in stress drop or other source properties, and appear in some cases to have systematic regional variations (e.g., Ekstrom and Dziewonski, 1988). This is why it is
better whenever possible to compute moment directly from the lowest frequency
part of the seismic spectra rather than to use magnitude measurements as a proxy for
moment.
From (9.73), a one unit increase in MW corresponds to a 103/2 ≃ 32 times
increase in moment. As we saw in Section 9.6, the average radiated seismic energy
is approximately proportional to moment so this means that seismic energy also
goes up by a factor of 32. Thus, on average a MS = 7 earthquake releases about 32
times more energy than a MS = 6 event and 1000 times more than a MS = 5 event.
This is consistent with the classic empirical Gutenberg-Richter relation between
ER and MS
log10 ER (joules) ≃ 4.8 + 1.5MS .
(9.74)
9.7 E A R T H Q U A K E M A G N I T U D E
287
Table 9.1: Some big earthquakes (M 0 in 1020 N m)
Date
Region
1960 May 22
MS
MW
M0
Chile
8.3
9.5
2000
1964 March 28
Alaska
8.4
9.2
820
2004 Dec 26
Sumatra-Andaman
8.5
9.1
680
1957 March 9
Aleutian Islands
8.2
9.1
585
1965 Feb 4
Aleutian Islands
8.7
140
2005 March 28
Sumatra
7.2
8.4
8.6
105
1977 Aug 19
Indonesia
7.0
7.9
8.3
36
2003 Sept 25
Hokkaido, Japan
6.9
8.1
8.3
31
1994 Oct 4
Shitokan, Kuriles
7.4
8.1
8.2
30
1994 June 9
Bolivia (deep)
6.9
8.2
26
2004 Dec 23
Macquarie Ridge
6.5
7.7
8.1
16
1989 May 23
Macquarie Ridge
6.4
8.2
8.2
20
1985 Sept 19
Michoacan, Mexico
6.5
8.3
8.0
14
1906 April 18
San Francisco
8.2
7.9
10
2008 May 12
Eastern Sichuan
6.9
8.0
7.9
9
2002 Nov 3
Denali, Alaska
7.0
8.5
7.8
7
2001 Nov 14
Kokoxili, Kunlun
6.1
8.0
7.8
6
1992 June 28
Landers, California
6.2
7.6
7.5
2
mb
6.2
However, this agreement is not really a coincidence because this equation was one
of the contributing reasons for the 2/3 factor in the definition of MW (Kanamori,
1977; Hanks and Kanamori, 1979). If we substitute MW from (9.73) into this
equation, we can obtain ER /M0 ≃ 6 × 10−5 , in rough agreement with the average
ẽ values plotted in Figure 9.21, at least over the 6 < MW < 8 interval in which
MW ≃ MS . However, it should be emphasized that the radiated energy ER is
best obtained through direct observations; the Gutenberg–Richter relation (9.74)
provides a crude estimate, but it can be in error by more than an order of magnitude
for individual events.
Table 9.1 lists some of the biggest earthquakes that have been recorded seismically, as well as some smaller strike–slip events for comparison purposes, and
Figure 9.26 plots global CMT results between 1976 and 2005, scaled by moment.
The listed mb and MS values show some of the effects of magnitude saturation. The
largest earthquakes of MW 8.5 and greater occur in subduction zones where the fault
area (= length × width) can be very large. Examples include the 1960 Chile earthquake, the 1964 Alaska earthquake, and the 2004 Sumatra-Andaman earthquake.
9. E A R T H Q U A K E S A N D S O U R C E T H E O R Y
288
2002 Denali
2001
Kokoxili
1986 Andreanof
1994 Shitokan
2003 Hokkaido
1995 Colima
1985 Michoacan
1996 Biak
1979 Colombia
2000 New Ireland
1994 Bolivia
2004
Sumatra
1977 Indonesia
2001 Peru
1977 Tonga
1995 Antofagasta
2005 Sumatra
1985 Valparaiso
2004 Macquarie
1989 Macquarie
1998 Antarctic Plate
Figure 9.26 The largest earthquakes from 1976 to 2005, with the focal mechanism area
proportional to seismic moment, as estimated from the Global CMT catalog.
In contrast, crustal strike–slip earthquakes generally do not exceed about MW = 8
because their fault widths are limited to the upper crust. Recent examples of large
strike–slip earthquakes include the 1989 and 2004 Macquarie Ridge earthquakes,
the 1998 Antarctic Plate earthquake, the 2001 Kokoxili (China) earthquake, and
the 2002 Denali (Alaska) earthquake. Most very large earthquakes are shallow; a
notable exception was the 1994 Bolivian earthquake at 630 km depth.
9.7.1 The b value
Smaller earthquakes occur much more frequently than large earthquakes. This trend
may be quantified in terms of a magnitude–frequency relationship. Gutenberg and
Richter noted that this relationship appears to obey a power law3 and obtained the
empirical formula
log10 N = a − bM,
3 A power-law distribution for earthquake energy had earlier been suggested by K. Wadati.
(9.75)
9.7 E A R T H Q U A K E M A G N I T U D E
289
where N is the number of events with magnitudes greater than or equal to M. In
this equation, a describes the total number of earthquakes, while the parameter b is
called the b-value and measures the relative number of large quakes compared to
small quakes. The b-value is generally found to lie between 0.8 and 1.2 for a wide
variety of regions and different magnitude scales (for a review, see Utsu, 2002a).
As we will discuss in Chapter 10, the b value is often used to estimate the fractal
dimension of fault systems (e.g., Aki, 1981; Turcotte, 1997). At b = 1 the number
of earthquakes increases by a factor of 10 for every unit drop in magnitude. For
example, if there is 1 M = 6 events per year in a region then we should expect
about 10 M = 5 events per year, 100 M = 4 events, etc.
Figure 9.27 shows N(MW ) computed for the global CMT catalog from 1976 to
2005. Between MW values 5.5 and 7.5, the distribution is well fit with b = 1. At
smaller magnitudes the increase in N drops off because earthquakes below about
MW = 5 are too small to be well recorded by the global seismic networks. Plots
like this are often used to evaluate catalog completeness – the lowest magnitude
to which a network or catalog includes all of the earthquakes. For MW > 7.5, the
numbers also drop below the b = 1 line, which may represent a change in the power
law or a temporary deficit in the number of very large global earthquakes. Because
moment increases by ∼30 for every unit increase in MW , while the number of events
only decreases by a factor ∼10, the total moment release from all of the seismicity
in a region is dominated by the largest events, rather than the accumulated sum
of many smaller events. Fortunately for humanity, (9.75) cannot remain valid for
Number of earthquakes
100 000
10 000
1000
100
b=1
10
1
5
6
7
8
9
MW
Figure 9.27 N(MW ) for the global CMT catalog from 1976 to 2005, compared to the predictions
of a power law decay with b value of 1.
290
9. E A R T H Q U A K E S A N D S O U R C E T H E O R Y
Table 9.2: The modified Mercalli scale, adapted from the abridged version in Bolt (1993).
I
II
III
IV
V
VI
VII
VIII
IX
X
XI
XII
Not felt except by a few under especially favorable circumstances.
Felt only by a few persons at rest, especially on upper floors of buildings. Delicately suspended
objects may swing.
Felt quite noticeably indoors, especially on upper floors of buildings, but many people do not
recognize it as an earthquake. Parked cars may rock slightly.Vibration like passing of truck. Duration
can be estimated.
During the day felt indoors by many, outdoors by few. At night some awakened. Dishes, windows,
doors disturbed, walls make creaking noise. Parked cars rocked noticeably. (0.015–0.02 g)
Felt by nearly everyone, many awakened. Some dishes, windows, etc., broken; a few instances of
cracked plaster; unstable objects overturned. Disturbance of trees, poles and other tall objects
sometimes noticed. (0.03–0.04 g)
Felt by all; many frightened and run outdoors. Some heavy furniture moved, a few instances of
fallen plaster or damaged chimneys. Damage slight. (0.06–0.07 g)
Everybody runs outdoors. Damage negligible in buildings of good design and construction; slightto-moderate damage in well-built ordinary structures; considerable in poorly built or badly designed structures; some chimneys broken. Noticed by people driving cars. (0.10–0.15 g)
Damage slight in specially designed structures; considerable in ordinary substantial buildings, with
partial collapse; great in poorly built structures. Panel walls thrown out of frame structures. Fall of
chimneys, factory stacks, columns, monuments, walls. Heavy furniture overturned. Sand and mud
ejected in small amounts. Changes in well water. Disturbs people driving cars. (0.25–0.30 g)
Damage considerable in specially designed structures; well-designed frame structures thrown out
of plumb; great in substantial buildings, with partial collapse. Buildings shifted off foundations.
Ground cracked conspicuously. Underground pipes broken. (0.5–0.55 g)
Some well-built wooden structures destroyed; with foundations; ground badly cracked. Rails bent.
Landslides considerable from river banks and steep slopes. Shifted sand and mud. Water splashed
over banks. (> 0.6 g)
Few, if any, masonry structures remain standing. Bridges destroyed. Broad fissures in ground. Underground pipelines completely out of service. Earth slumps and land slips in soft ground. Rails
bent greatly.
Damage total. Waves seen on ground surfaces. Lines of sight and level distorted. Objects thrown
upward into the air.
arbitrarily large earthquakes because the finite extent of Earth’s faults means there
is a maximum possible earthquake size. We need not fear a MW = 11 earthquake
every 1000 years.
9.7.2 The intensity scale
Another measure of the earthquake strength is the seismic intensity, which describes
the local strength of ground shaking as determined by damage to structures and the
perceptions of people who experienced the earthquake. The intensity scale most
often used today in the United States is the modified Mercalli scale, in which
intensity ranges from I to XII (Roman numerals). As shown in Table 9.2, a value
of I indicates shaking that is felt only by a few people, V is felt by almost everyone, VIII causes great damage in poorly built structures, and XII indicates total
destruction. Although approximate peak accelerations can be assigned to these levels, the great advantage of the Mercalli scale is that it can be used to examine
9.8 F I N I T E S L I P M O D E L I N G
291
historic earthquakes that were not recorded by modern instruments. This is often
done by interviewing witnesses and studying old newspaper accounts. Once the
intensity has been estimated at a number of different sites, a contour map of the
intensities can be constructed. The earthquake location is then identified from this
map as the spot of maximum intensity, and an approximate magnitude can be estimated from the area surrounded by the different intensity contours. This technique
provides the only practical way to obtain probable locations and magnitudes for
many older events. The importance of these estimates is illustrated by the seismicity in the eastern United States, where no large earthquake was recorded in the
twentieth century. However, several large events occurred in the nineteenth century, including a series of three large earthquakes that struck in 1811–1812 near
New Madrid along the Mississippi River in Missouri. By studying and mapping
accounts of these events, which were felt across much of the eastern United States,
it is possible to constrain their sizes and locations and make estimates of the impact
of a future earthquake in the same area. For many years the New Madrid events
were thought to be about magnitude 8, but a recent reanalysis of intensity data by
Hough et al. (2000) indicates that they were more likely MW 7 to 7.5.
A modern way to quickly estimate earthquake intensities is the “Did You Feel It?’’
website maintained by the United States Geological Survey (USGS), in which users
are asked to answer a short series of questions about how strongly they felt the shaking and to identify their location (http://earthquake.usgs.gov/eqcenter/dyfi.php).
These responses are then compiled into an intensity map that can provide surprising detail, as shown in Figure 9.28 for the 2003 M 6.5 San Simeon earthquake in
central California. Spatial resolution in the United States is currently limited to zip
code boundaries, but in the future it is likely that more precise user locations will
make possible more detailed maps.
9.8 Finite slip modeling
Up to this point, we have described seismic sources with a limited number of
parameters, such as the moment tensor or focal mechanism, the scalar moment, the
duration or corner frequency, the rupture velocity and direction, the stress drop,
and the radiated energy. These represent averages over the source region and for
small earthquakes this is most of what we can hope to learn because of the limited
bandwidth of far-field seismic records. However, for large earthquakes it is often
possible to invert for more detailed source properties because it it possible to separately resolve the seismic radiation from different parts of the fault. A common
approach is to discretize the fault into a series of rectangular cells, solve for the
Green’s function that gives the response at each of the available seismic stations,
9. E A R T H Q U A K E S A N D S O U R C E T H E O R Y
292
MERCED
BIG PINE
37°N
HOLLISTER
INDEPENC
FRESNO
MOKTEREY
VISALLA
KINGCITY
COALINGA
PORTERVILLE
36°N
PARKFIELD
SAN SIMEON
l.ISABELLA
ASO ROBLES
BAKERSFIELD
SAN LUIS OBISPO
MCLA
35°N
SANTA MARIA
LOMPOC
km
0
40
80
POINT CONCEPTIONSANTABARBARAOLAI
17204 responses in 1057 ZIP areas. Max intensity. VIII
121°W
122°W
VENTURA
120°W
V
INTENSITY
I
II-III
IV
SHAKING
Nolid1
Weak
Light
Moderate Strong
DAMAGE
none
none
name
Very light
VI
Light
GORMAN
L.HUGHESCA
PALM
eastaic
NORTHRIDGE
PASAL
119°W
VII VIII
Very strong
VIII
IX
X+
Severe
viden1
Extreme
Moderate Moderate/Heavy
Heavy Very Heavy
Figure 9.28 Ground shaking intensities for the 2003 San Simeon earthquake in California, as
measured from 17 204 responses to the USGS ‘‘Did You Feel It?’’ website.
and then set up an inverse problem using the superposition principle to solve for
the time-dependent slip distribution that best fits all of the data. If geodetic data are
also available, these can also be included in the inversion to provide constraints on
the total slip. As in the tomographic inversions for 3-D velocity structure discussed
in Chapter 5, the inverse problem often is stabilized by applying regularization constraints, such as only permitting slip in one direction and solving for the smoothest
or the minimum slip solution.
The resulting finite slip models typically show that large earthquakes have irregular slip distributions and that their total moment is dominated by one or two large
slip regions where the stress drop is much higher than surrounding parts of the fault.
Rupture velocity is also not always constant during earthquakes. As an example,
Figure 9.29 shows the Custodio et al. (2005) model of the 2004 Parkfield earthquake obtained from an inversion of the strong motion data. From the hypocenter
9.9 T H E H E A T F L O W P A R A D O X
0.26
0
293
0.52
Slip (m)
NW
SE
Depth (km)
0
5
10
15
0
10
20
30
40
Distance (km)
Figure 9.29 The Custodio et al. (2005) slip model for the 2004 Parkfield earthquake.The star
shows the hypocenter location.The contours show the slip onset times at 1 s intervals.
the rupture propagated northwest along the San Andreas Fault for about 10 s. There
were two major areas of slip, one near the hypocenter and the other 10 to 20 km to
the northwest at a depth between 2 and 8 km.
Finite slip models provide detailed constraints on rupture dynamics that are
valuable for understanding earthquake physics. Results so far point to highly
heterogeneous faulting processes with large variations in moment release and stress
drop. But there are problems with non-uniqueness in many of the inversions and
results from different groups for the same earthquake do not always agree very
well. As inversions move to higher frequencies, propagation path effects on strong
motion data become increasingly important and link the rupture inversion problem
to the 3-D velocity inversion problem. This is an active area of research and eventually improved modeling methods and denser seismic and geodetic networks will
help to provide clearer images of fault rupture.
9.9 The heat flow paradox
As discussed in Section 9.6.1, the frictional and fracture energy release during
faulting is given by
EF = σf DA
(9.76)